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The Tits Alternative for Out(Fn) II: a Kolchin Type Theorem

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The Tits Alternative for Out(Fn) II: a Kolchin Type Theorem Annals of Mathematics, 161 (2005), 1–59 The Tits alternative for Out(Fn) II: A Kolchin type theorem By Mladen Bestvina, Mark Feighn, and Michael Handel* Abstract This is the second of two papers in which we prove the Tits alternative for Out(Fn). Contents 1. Introduction and outline 2. Fn-trees 2.1. Real trees 2.2. Real Fn-trees 2.3. Very small trees 2.4. Spaces of real Fn-trees 2.5. Bounded cancellation constants 2.6. Real graphs 2.7. Models and normal forms for simplicial Fn-trees 2.8. Free factor systems 3. Unipotent polynomially growing outer automorphisms 3.1. Unipotent linear maps 3.2. Topological representatives 3.3. Relative train tracks and automorphisms of polynomial growth 3.4. Unipotent representatives and UPG automorphisms 4. The dynamics of unipotent automorphisms 4.1. Polynomial sequences 4.2. Explicit limits 4.3. Primitive subgroups 4.4. Unipotent automorphisms and trees 5. A Kolchin theorem for unipotent automorphisms 5.1. F contains the suffixes of all nonlinear edges 5.2. Bouncing sequences stop growing 5.3. Bouncing sequences never grow *The authors gratefully acknowledge the support of the National Science Foundation. 2 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL 5.4. Finding Nielsen pairs 5.5. Distances between the vertices 5.6. Proof of Theorem 5.1 6. Proof of the main theorem References 1. Introduction and outline Recent years have seen a development of the theory for Out(Fn), the outer automorphism group of the free group Fn of rank n, that is modeled on Nielsen- Thurston theory for surface homeomorphisms. As mapping classes have either exponential or linear growth rates, so free group outer automorphisms have either exponential or polynomial growth rates. (The degree of the polynomial can be any integer between 1 and n−1; see [BH92].) In [BFH00], we considered individual automorphisms with primary emphasis on those with exponential growth rates. In this paper, we focus on subgroups of Out(Fn) all of whose elements have polynomial growth rates. To remove certain technicalities arising from finite order phenomena, we restrict our attention to those outer automorphisms of polynomial growth ∼ n whose induced automorphism of H1(Fn; Z) = Z is unipotent. We say that such an outer automorphism is unipotent. The subset of unipotent outer auto- morphisms of Fn is denoted UPG(Fn) (or just UPG). A subgroup of Out(Fn) is unipotent if each element is unipotent. We prove (Proposition 3.5) that any polynomially growing outer automorphism that acts trivially in Z/3Z- homology is unipotent. Thus every subgroup of polynomially growing outer automorphisms has a finite index unipotent subgroup. The archetype for the main theorem of this paper comes from linear groups. A linear map is unipotent if and only if it has a basis with respect to which it is upper triangular with 1’s on the diagonal. A celebrated theorem of Kolchin [Ser92] states that for any group of unipotent linear maps there is a basis with respect to which all elements of the group are upper triangular with 1’s on the diagonal. There is an analogous result for mapping class groups. We say that a map- ping class is unipotent if it has linear growth and if the induced linear map on first homology is unipotent. The Thurston classification theorem implies that a mapping class is unipotent if and only if it is represented by a composition of Dehn twists in disjoint simple closed curves. Moreover, if a pair of unipotent mapping classes belongs to a unipotent subgroup, then their twisting curves cannot have transverse intersections (see for example [BLM83]). Thus every unipotent mapping class subgroup has a characteristic set of disjoint simple closed curves and each element of the subgroup is a composition of Dehn twists along these curves. THE TITS ALTERNATIVE FOR Out(Fn) II 3 Our main theorem is the analogue of Kolchin’s theorem for Out(Fn). Fix once-and-for-all a wedge Rosen of n circles and permanently identify its fun- damental group with Fn. A marked graph (of rank n) is a graph equipped with a homotopy equivalence from Rosen; see [CV86]. A homotopy equiva- lence f : G → G on a marked graph G induces an outer automorphism of the fundamental group of G and therefore an element O of Out(Fn); we say that f : G → G is a representative of O. Suppose that G is a marked graph and that ∅ = G0 G1 ··· GK = G is a filtration of G where Gi is obtained from Gi−1 by adding a single edge Ei. A homotopy equivalence f : G → G is upper triangular with respect to the filtration if each f(Ei)=viEiui (as edge paths) where ui and vi are closed paths in Gi−1. If the choice of filtration is clear then we simply say that f : G → G is upper triangular. We refer to the ui’s and vi’s as suffixes and prefixes respectively. An outer automorphism is unipotent if and only if it has a representative that is upper triangular with respect to some filtered marked graph G (see Section 3). For any filtered marked graph G, let Q be the set of upper triangular homotopy equivalences of G up to homotopy relative to the vertices of G.By Lemma 6.1, Q is a group under the operation induced by composition. There is a natural map from Q to UPG(Fn). We say that a unipotent subgroup of Out(Fn)isfiltered if it lifts to a subgroup of Q for some filtered marked graph. We denote the conjugacy class of a free factor F i by [[F i]]. If F 1 ∗F 2 ∗···∗ F k is a free factor, then we say that the collection F = {[[F 1]], [[F 2]],...,[[F k]]} is a free factor system. There is a natural action of Out(Fn) on free factor systems and we say that F is H-invariant if each element of the subgroup H fixes F. A (not necessarily connected) subgraph K of a marked real graph determines a free factor system F(K). A partial order on free factor systems is defined in subsection 2.8. We can now state our main theorem. Theorem 1.1 (Kolchin theorem for Out(Fn)). Every finitely generated unipotent subgroup H of Out(Fn) is filtered. For any H-invariant free factor system F, the marked filtered graph G can be chosen so that F(Gr)=F for some filtration element Gr. The number of edges of G can be taken to be 3n − bounded by 2 1 for n>1. It is an interesting question whether or not the requirement that H be finitely generated is necessary or just an artifact of our proof. Question. Is every unipotent subgroup of Out(Fn) contained in a finitely generated unipotent subgroup? 4 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL Remark 1.2. In contrast to unipotent mapping class subgroups which are all finitely generated and abelian, unipotent subgroups of Out(Fn) can be quite large. For example, if G is a wedge of n circles, then a filtration on G corre- sponds to an ordered basis {e1,...,en} of Fn and elements of Q correspond to automorphisms of the form ei → aieibi with ai,bi ∈e1,...,ei−1. When n>2, the image of Q in UPG(Fn) contains a product of nonabelian free groups. This is the second of two papers in which we establish the Tits alternative for Out(Fn). Theorem (The Tits alternative for Out(Fn)). Let H be any subgroup of Out(Fn). Then either H is virtually solvable, or contains a nonabelian free group. For a proof of a special (generic) case, see [BFH97a]. The following corollary of Theorem 1.1 gives another special case of the Tits alternative for Out(Fn). The corollary is then used to prove the full Tits alternative. Corollary 1.3. Every unipotent subgroup H of Out(Fn) either contains a nonabelian free group or is solvable. Proof. We first prove that if Q is defined as above with respect to a marked filtered graph G, then every subgroup Z of Q either contains a nonabelian free group or is solvable. Let i ≥ 0 be the largest parameter value for which every element of Z restricts to the identity on Gi−1.Ifi = K + 1, then Z is the trivial group and we are done. Suppose then that i ≤ K. By construction, each element of Z satisfies Ei → viEiui where vi and ui are paths (that depend on the element of Z)inGi−1 and are therefore fixed by every element of Z. The suffix map S : Z→Fn, which assigns the suffix ui to the element of Z, is therefore a homomorphism. The prefix map P : Z→Fn, which assigns the inverse of vi to the element of Z, is also a homomorphism. If the image of P×S: Z→Fn × Fn contains a nonabelian free group, then so does Z and we are done. If the image of P×Sis abelian then, since Z is an abelian extension of the kernel of P×S, it suffices to show that the kernel of P×Sis either solvable or contains a nonabelian free group. Upward induction on i now completes the proof. In fact, this argument shows that Z 3n − is polycyclic and that the length of the derived series is bounded by 2 1 for n>1. For H finitely generated the corollary now follows from Theorem 1.1. When H is not finitely generated, it can be represented as the increasing union of finitely generated subgroups.
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